Mutual Fund Mastery puzzles, solved step by step

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All topicsCompounding and time value9Statistics, correlation and diversification8Bond maths and duration10Performance measurement and returns8Costs and fee drag8Valuation riddles11Logic and numeracy brainteasers6Estimation and market sizing7Probability and expected value8NAV, units and fund mechanics7Risk, volatility and drawdown8Behavioural traps6Withdrawals and after-tax arithmetic4
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  1. 048A fund's NAV at the end of each quarter runs 100, 130, 110, 140, 91, 120. What is its maximum drawdown, and why is it not measured from the starting NAV of 100?Risk, volatility and drawdownWarm upRisk and complianceIndian AMCs

    Try it first

    What is the maximum drawdown?

    Show the worked solution

    The maximum drawdown is 35%, from the peak of 140 to the trough of 91. Drawdown is always measured from the highest value reached before the fall, because that is the value an investor held and then lost. Measured from the start, 91 looks like a mild 9% dip, which hides how much was lost on the way. The fund later climbs to 120, still 14.3% below its peak.

    Why measure from the peak and not from the start?

    A house bought for Rs 1 crore, valued at Rs 1.4 crore at the top of a boom and then sold for Rs 91 lakh, did not lose 9%. Its owner watched Rs 49 lakh disappear. A drawdown measures the pain of falling from the best point reached, so it runs from the running peak, which keeps rising whenever the NAV sets a new high. Anyone who invested at 140, or held through it, lost 35%, and that is the risk the measure is built to show.

    Drawdown is measured from the highest point reached, not from the start80100120140t0t1t2t3t4t5running peak100130110140 peak91 trough120-35%maximum drawdown-15.4%from the start: only 9%
    The NAV path touches 140 and then falls to 91, a 35% drawdown from the running peak, while the earlier dip from 130 to 110 is only 15.4%; measured from the starting 100, the low of 91 would look like just a 9% fall.

    How do you compute it step by step?

    Walk along the path and keep two numbers: the highest NAV so far and the current NAV. At each point the drawdown is current over peak, less one. The maximum drawdown is the worst of those readings, here 91 over 140 less one, minus 35%. The dip from 130 to 110 gives minus 15.4%, a smaller drawdown. The final 120 against a peak of 140 is a drawdown of minus 14.3% still open.

    The relationship
    DDt=NAVtmax⁡s≤tNAVs−1,MDD=min⁡tDDt=91140−1=−35%DD_t = \frac{NAV_t}{\max_{s \le t} NAV_s} - 1, \qquad MDD = \min_t DD_t = \frac{91}{140} - 1 = -35\%
    NAV_tthe NAV at time t
    max NAV_sthe running peak up to time t
    MDDthe maximum drawdown, the worst reading
    What it says in wordsEach drawdown compares today's NAV with the best NAV so far; the maximum drawdown is the deepest of them.

    Add the recovery maths: from 91 the fund needs a 53.8% rise just to get back to 140. And the measure depends on how often you look. These are quarter-end NAVs; a daily series could show a deeper trough between the quarter ends, so a maximum drawdown should always be quoted with its data frequency and period.

    Where candidates lose it

    The trap is measuring from the starting NAV and answering 9%, because the question opens with 100. The interviewer is checking that you know the reference point moves up with every new high.

    The second slip is dividing by the trough, 49 over 91, and answering 54%. That is the gain needed to recover, not the drawdown. Say both numbers and label them.

    What the interviewer asks next

    • What gain does the fund need from 120 to set a new high?
    • Why might a daily NAV series show a larger maximum drawdown than quarter-end NAVs?
    • Two funds have the same volatility; one has a much larger maximum drawdown. What could explain it?
  2. 057A fund cuts its expense ratio from 1.5% to 1.2%. Is that a 0.3% cut, a 30 basis point cut or a 20% cut, and which description will a client misunderstand?Logic and numeracy brainteasersWarm upIndian AMCsGlobal asset managers

    Try it first

    Which descriptions of the change are correct?

    Show the worked solution

    It is a 0.3 percentage point cut, which is 30 basis points, and a 20% relative cut in the fee; "a 0.3% cut" is the sloppy one. Basis points remove the ambiguity. The 20% is true but invites a client to imagine returns rising 20%; on Rs 10 lakh the saving is Rs 3,000 a year, and a net return of 8.5% becomes 8.8%.

    Why are there two different answers that are both right?

    A shop raises the price of milk from Rs 50 to Rs 60. It went up by Rs 10, and it went up by 20%. Nobody confuses those, because rupees and percent look different. With rates, the change and the base are both percentages, so "percent" can mean the gap between two rates or the change relative to the old rate, and you must say which. The gap between 1.5% and 1.2% is 0.3 percentage pointsThe plain difference between two percentages. From 1.5% to 1.2% is a fall of 0.3 percentage points.. Relative to the old fee, 0.3 is a fifth of 1.5, a 20% cut.

    One fee cut, three honest names, one misleading onePercentage points0.3 pp1.5% minus 1.2%The difference of two ratesBasis points30 bps1 bp = 0.01 pointThe desk's unit; no ambiguityRelative change-20%0.3 / 1.5Of the fee, not of returnsSaid as "a 0.3% cut"Heard as 0.3% of the fee:1.5% x 0.997 = 1.4955%A cut 100 times too smallWhat it means on Rs 10 lakhFee Rs 15,000 falls to Rs 12,000: Rs 3,000 savedNet return at a 10% gross: 8.5% to 8.8%Return up 3.5% in relative terms, not 20%
    The same fee cut from 1.5% to 1.2% is 0.3 percentage points, 30 basis points or a 20% relative cut, and on Rs 10 lakh it saves Rs 3,000 a year while lifting a net return from 8.5% to 8.8%.

    Which description will a client get wrong, and what should you say instead?

    "A 0.3% cut" fails one way: a careful reader can take it as 0.3% of the fee, which would leave 1.5% x 0.997 = 1.4955%, a cut a hundred times too small. "A 20% cut" fails the other way: it is true of the fee, but a client hears 20% and imagines the investment doing 20% better. The fee is a slice of a slice; a 20% smaller slice lifts a net return of 8.5% to 8.8%, only 3.5% better in relative terms. The clean form for a desk is basis pointsHundredths of a percentage point. 30 basis points equal 0.30 percentage points.; the clean form for a client is rupees: Rs 15,000 a year on Rs 10 lakh becomes Rs 12,000.

    The relationship
    1.5%−1.2%=0.3 pp=30 bps1.5−1.21.5=20%1.5\% - 1.2\% = 0.3\text{ pp} = 30\text{ bps} \qquad \frac{1.5 - 1.2}{1.5} = 20\%
    pppercentage points, the plain gap between two rates
    bpsbasis points, one hundredth of a percentage point
    1.5the old fee, the base for the relative change
    What it says in wordsSubtract for points, divide by the old rate for percent change, and name which one you mean.

    The same confusion runs through everything a fund desk reports: a yield moving from 7.0% to 7.5% is 50 basis points, not 0.5%, and a fund outperforming by 2% might mean two points or a fiftieth more. The limit worth stating: neither unit tells the client the money, so pair any rate change with the rupee effect on their actual holding.

    Where candidates lose it

    The trap is picking one answer and defending it. The interviewer is not after 0.3 or 20; they want to hear that these are two different quantities and that the unit must be named every time.

    The second miss is not taking the client's view. Saying 20% to a client is legally true and practically misleading, because it invites them to apply the 20% to their returns. Translate into rupees on their holding.

    What the interviewer asks next

    • A bond yield rises from 6.8% to 7.2%. Describe the change in three correct ways.
    • A fund's return beat the index's 10% by 2%. List the two things this could mean.
    • Why do desks quote spreads in basis points rather than percent?
  3. 069A 1.5% expense ratio sounds small. If an equity fund's expected return before fees is 10%, what fraction of the return goes in fees? For a liquid fund returning 6.5% before a 0.25% expense ratio, what fraction?Costs and fee dragWarm upIndian AMCsDistribution and sales

    Try it first

    What share of the equity fund's expected return does the fee take?

    Show the worked solution

    The equity fund's fee takes 15% of its expected return; the liquid fund's takes about 3.8%. A fee is charged on the money invested, but it comes out of the return, so it should be measured against the return. 1.5 / 10 is 15%; 0.25 / 6.5 is 3.85%. Measured against the 3.5% extra that equity is expected to earn over cash, the 1.5% fee takes 43%.

    Why does 1.5% sound smaller than it is?

    An agent who charges you 1.5% of your house price to sell it sounds cheap until you remember you only made Rs 10 lakh of profit on a Rs 1 crore house: the Rs 1.5 lakh fee takes 15% of the profit. Fund fees are quoted on the money invested, but they are paid out of the return, so the honest size of a fee is the fee divided by the return it takes from. For the equity fund, 1.5 / 10 = 15%: almost one rupee in every seven the fund is expected to earn goes to costs. For the liquid fund, 0.25 / 6.5 = 3.85%.

    Judge a fee against the return it takes fromEquity fundkept 8.50%fee 1.5%15% of the returnLiquid fundkept 6.25%fee 0.25%3.8% of the returnEquity fund, extra return over cash onlykept 2.00%fee 1.5%43% of the extraBar length = return in % a year (red = fee). The same 1.5% weighs far more against the 3.5% extra that equity is held for.
    The equity fund's 1.5% fee takes 15% of a 10% expected return, the liquid fund's 0.25% fee takes 3.8% of 6.5%, and against the 3.5% extra return that equity is held for, the same 1.5% takes 43%.

    What is the sharper way to measure the equity fee?

    An investor holds equity instead of cash to earn the extra return, the equity risk premiumThe return equities are expected to earn above cash or short government bills, as pay for their extra risk.. If cash pays 6.5% and equity is expected to make 10%, the extra is 3.5%. A 1.5% fee takes 1.5 / 3.5 = 43% of that extra, so close to half of the reason for owning equity goes in costs. That is why low-cost index funds draw so much money: the fee matters most where the expected extra return is thin.

    The relationship
    feeexpected return=1.510=15%0.256.5=3.8%1.510−6.5=43%\frac{\text{fee}}{\text{expected return}} = \frac{1.5}{10} = 15\% \qquad \frac{0.25}{6.5} = 3.8\% \qquad \frac{1.5}{10 - 6.5} = 43\%
    feethe expense ratio, % of assets a year
    expected returnwhat the fund is expected to earn before fees, % a year
    6.5the cash rate, used to find the extra return
    What it says in wordsA fee's real size is its share of the return, and sharper still, its share of the extra return you took risk for.

    Two limits keep this honest. Expected returns are assumptions, not promises, and in a year when the fund falls the fee takes more than all of the return. And a higher fee can still be worth paying if the manager adds more than the fee in return after costs, which is rare enough that the evidence for it should be asked for, not assumed.

    Where candidates lose it

    The trap is comparing the fee with the investment, the way it is quoted, and calling it small. The interviewer wants the fee compared with the return, which makes 1.5% look ten times larger.

    The second miss is treating all fees alike. A 0.25% fee on a liquid fund and a 1.5% fee on an equity fund take very different shares of what each is expected to earn; say both shares and the comparison makes itself.

    What the interviewer asks next

    • Inflation is 5%. What share of the equity fund's real return does the fee take?
    • A fund charges 2% and its manager is expected to beat the index by 1% before fees. What is the investor's expected result against the index?
    • Why do fee differences matter more in debt funds than they first appear to?
  4. 070Two people each invest Rs 10,000 a month at 12% a year until they are 60, one starting at 25 and one at 35. The early starter puts in only Rs 12 lakh more, yet ends with about 3.4 times the corpus. Why?Compounding and time valueWarm upIndian AMCsDistribution and sales

    Try it first

    Of the early starter's corpus at 60, roughly how much comes from the first ten years of saving alone?

    Show the worked solution

    Because the extra ten years come first, and the earliest rupees compound the longest. At 1% a month, the early starter reaches about Rs 6.50 crore and the late starter about Rs 1.90 crore, 3.4 times less, though the gap in money paid in is only Rs 12 lakh. That Rs 12 lakh, saved from 25 to 35, grows to about Rs 4.60 crore by 60, 71% of the early starter's corpus.

    Why is the early starter so far ahead for so little extra?

    Plant a mango sapling at 25 and another at 35, and at 60 the first tree is not slightly bigger; it has had ten more seasons of growth on a trunk that kept getting larger. Money saved early does not just add ten more years of deposits; it gives every rupee in those years twenty-five more years of compounding after them. The late starter pays in Rs 30 lakh over 25 years and ends with Rs 1.90 crore. The early starter pays Rs 42 lakh over 35 years and ends with Rs 6.50 crore.

    Rs 10,000 a month at 12%: start at 25 or at 35First ten yearsRs 12 lakh paid in01234567Rs crore2530354045505560ageStart at 25: Rs 6.50 croreRs 42 lakh paid inStart at 35: Rs 1.90 croreRs 30 lakh paid inRatio at 60: 3.4 timesfor only Rs 12 lakh more paid in
    Rs 10,000 a month at 12% grows to Rs 6.50 crore by 60 when started at 25 and to Rs 1.90 crore when started at 35, because the Rs 12 lakh saved in the first ten years alone grows to about Rs 4.60 crore.

    How much does the first decade contribute on its own?

    Split the early starter's plan in two. From 25 to 35 she saves Rs 12 lakh, which is about Rs 23.2 lakh by 35. From then she saves exactly what the late starter saves, so that part ends at Rs 1.90 crore. The Rs 23.2 lakh from the first decade compounds for 25 more years at 1% a month and becomes about Rs 4.60 crore, 71% of her final corpus. Her first ten years are worth more at 60 than the late starter's entire 25 years of saving.

    The relationship
    FV=10,000×(1.01)n−10.01×1.01n=420:Rs 6.50 crn=300:Rs 1.90 crFV = 10{,}000 \times \frac{(1.01)^n - 1}{0.01} \times 1.01 \qquad n = 420: \text{Rs }6.50\text{ cr} \qquad n = 300: \text{Rs }1.90\text{ cr}
    10,000the monthly investment, Rs
    0.01the monthly rate, 12% a year taken as 1% a month
    nthe number of monthly instalments, 420 from 25 and 300 from 35
    x 1.01each instalment is invested at the start of the month
    What it says in wordsThe future value of a monthly plan grows with the number of months as a power, not in a straight line, so extra months at the start count most.

    State the assumptions, because the size of the gap depends on them. A steady 12% every year is a simplification; real equity returns swing, and the order of good and bad years matters for a monthly saver. Inflation shrinks both corpora in today's rupees, though not the ratio. And the gap narrows if the late starter saves more each month; to match the early starter at 60 she would have to invest about 3.4 times as much, around Rs 34,000 a month.

    Where candidates lose it

    The trap is reasoning in straight lines: ten more years out of thirty-five, so about 40% more money. Compounding makes the earliest deposits the most valuable, not the least, and candidates who scale by years miss the point of the question.

    The second miss is quoting the corpus without the assumptions. Say it rests on a constant 12%, monthly compounding and deposits at the start of each month, and that real returns arrive unevenly.

    What the interviewer asks next

    • How much a month would the late starter need to invest to match Rs 6.5 crore at 60?
    • The early starter stops saving at 35 and never adds another rupee. What does she have at 60?
    • How does a 10% return instead of 12% change the ratio between the two?
  5. 071A stock trades at 20 times earnings and pays out 40% of its profit as dividends. What is its dividend yield?Valuation riddlesWarm upHoulihan LokeyChicago · 2026

    Try it first

    What is the dividend yield?

    Show the worked solution

    2%. Flip the P/E to get the earnings yield: a P/E of 20 means the company earns 1/20, or 5%, of its share price each year. It pays out 40% of those earnings, so the dividend is 40% of 5%, which is 2% of the price. A Rs 400 share earning Rs 20 pays Rs 8, and 8 / 400 is 2%.

    How do you get from a P/E to a yield?

    If a flat costs Rs 20 lakh and earns Rs 1 lakh a year in rent, it costs 20 years of rent, and the rent is 5% of the price. The same flip works for shares. A P/E of 20 means the share costs 20 years of earnings, so the earnings yieldEarnings per share divided by the share price: the P/E turned upside down. A P/E of 20 is an earnings yield of 5%. is 1/20 = 5%. Only part of those earnings reach the shareholder as cash. With a payout ratioThe share of profit a company pays out as dividends. The rest is retained and reinvested in the business. of 40%, the dividend is 0.4 x 5% = 2% of the price, and the other 3% stays in the company.

    Dividend yield = payout ratio x earnings yieldEarnings yield = 1 / P/E = 1 / 20Paid out: 2%Kept: 3%= 5%40% of earnings60% of earnings0%1%2%3%4%5%One share, in rupeesPriceRs 400EarningsRs 20Dividend (40%)Rs 8Dividend yield8 / 400 = 2%P/E = 400 / 20 = 20; the kept Rs 12 is the 3% the company reinvests
    A P/E of 20 is an earnings yield of 5%, and paying out 40% of it gives a dividend yield of 2% with 3% retained, as a Rs 400 share earning Rs 20 and paying Rs 8 shows.
    The relationship
    DP=DE×EP=0.40×120=2%\frac{D}{P} = \frac{D}{E} \times \frac{E}{P} = 0.40 \times \frac{1}{20} = 2\%
    D / Pdividend yield, dividend per share over price
    D / Epayout ratio, dividend per share over earnings per share
    E / Pearnings yield, the inverse of the P/E
    What it says in wordsDividend yield is the payout ratio times the earnings yield, because the earnings cancel out.

    What does the other 3% do, and where does this identity help?

    The retained 3% is not lost; it is reinvested. If the company earns 15% on what it reinvests, keeping 60% of profit lets it grow earnings by about 0.6 x 15% = 9% a year, and the dividend yield plus that growth, 11%, is a rough estimate of the shareholder's long-run return. That is the logic of a dividend discount model in one line. The identity also works backwards in an interview: a stock yielding 2% with a 40% payout must be on a P/E of 20.

    The limits: the P/E uses one year's earnings, which may be unusually high or low, and the payout ratio can change from year to year. Buybacks return cash too, so a company paying low dividends but buying back shares can return more than its dividend yield suggests. And the growth estimate assumes the company keeps earning 15% on new money, which gets harder as it grows.

    Where candidates lose it

    The trap is dividing the wrong things: 40% by 20 gives 2 by luck, but candidates who do it cannot explain why and fall over on the follow-up. Others turn 20 into 20% or forget to flip the P/E at all.

    Say the identity out loud before the number: dividend yield equals payout ratio times earnings yield. Then the arithmetic is one line and every variation of the question is the same line.

    What the interviewer asks next

    • A stock yields 3% and pays out 60% of earnings. What is its P/E?
    • The company raises its payout to 80% with no change in price. What happens to the yield and to future growth?
    • Why might a fund manager prefer a 1% yielder that buys back shares to a 3% yielder that does not?

    Asked at Houlihan Lokey, Private Funds Advisory, Chicago, 2026 (Wall Street Oasis): Mostly technical, with standard accounting and valuation ratio questions.

  6. 075A bond with a 7% annual coupon trades at 104. Is its yield to maturity above or below 7%, and what is its current yield?Bond maths and durationWarm upVanguardMalvern · 2023

    Try it first

    Which ordering is right for this bond?

    Show the worked solution

    The yield to maturity is below 7%, and the current yield is 6.73%. Current yield is the coupon over the price: 7 / 104 = 6.73%. The yield to maturity is lower still, because you pay 104 and get back only 100, so the Rs 4 premium is a loss spread over the bond's life. For an assumed five years left, the yield to maturity is about 6.05%.

    Why must the yield be below the coupon when the price is above 100?

    Pay Rs 104 for a gift voucher worth Rs 100 that also pays Rs 7 of cashback a year. The cashback is generous, but you have overpaid for the voucher by Rs 4. A bond priced above par returns less than its coupon, because the buyer pays more than the Rs 100 that comes back at maturity, and that premium is lost along the way. The coupon of 7% is set on the face value of 100 and never changes. The current yieldThe yearly coupon divided by the bond price today. It ignores any gain or loss as the price moves to face value at maturity. divides the same Rs 7 by what you actually pay: 7 / 104 = 6.73%.

    Above par, the yields sit below the coupon0%1%2%3%4%5%6%7%7.00%Coupon rate7 / 100 face6.73%Current yield7 / 104 price6.05%Yield to maturityadds the Rs 4 lossPrice pulls to parpar 100104 todayyr 0yr 1yr 2yr 3yr 4yr 5Rs 4 premium lost by maturity
    A 7% coupon bond bought at 104 has a current yield of 6.73% and, with five years left, a yield to maturity of 6.05%, because its price pulls down to 100 by maturity and the Rs 4 premium is lost.

    How far below 7% is the yield to maturity?

    That depends on how long the bond has to run, which the question does not say, so name an assumption. With five years left, the Rs 4 premium is lost at roughly Rs 0.80 a year. A quick estimate takes the coupon less that yearly loss, Rs 6.20, over the average of the purchase and redemption prices, 102: about 6.08%. The exact yield to maturityThe single discount rate that makes all remaining coupons and the final repayment worth exactly the price paid today. is 6.05%. The longer the bond, the thinner the yearly slice of the premium and the closer the yield to maturity sits to the current yield.

    The relationship
    104=∑t=157(1+y)t+100(1+y)5  ⇒  y≈6.05%104 = \sum_{t=1}^{5} \frac{7}{(1+y)^t} + \frac{100}{(1+y)^5} \;\Rightarrow\; y \approx 6.05\%
    104the price paid
    7the yearly coupon per 100 of face
    100the repayment at maturity
    ythe yield to maturity
    What it says in wordsThe yield to maturity counts the coupons and the loss of the premium, so it ends below both the coupon and the current yield.

    For a debt fund this ordering is everyday arithmetic: when rates fall, older high-coupon bonds trade above par, and the fund's quoted portfolio yield sits below the coupons it receives. The limit of yield to maturity: it assumes the bond is held to maturity, never defaults and that coupons are reinvested at the same yield, none of which is guaranteed. A bond callable before maturity at par makes the premium an even bigger risk.

    Where candidates lose it

    The trap is quoting 7% as the yield because that is the coupon. The coupon is fixed on the face value; the yield depends on the price you pay, and above par it is lower.

    The second miss is stopping at the current yield. Name the order, coupon above current yield above yield to maturity, and give the reason for the last step: the premium is lost by maturity.

    What the interviewer asks next

    • The same bond trades at 96. Put the coupon, current yield and yield to maturity in order.
    • With 20 years left instead of 5, is the yield to maturity closer to or further from the current yield?
    • Why do debt funds holding many premium bonds report a portfolio yield below their average coupon?

    Asked at Vanguard, Investments, Malvern, 2023 (Wall Street Oasis): Questions asked ranged from resume stuff, global macro/current news stuff, and simple bond math since I expressed interest in FICC.

  7. 076A market index has fallen five days in a row. Over its history it has risen on 52% of days, and each day's move is independent of the last. A client asks whether it is now due a rise. What is the chance the index rises tomorrow?Behavioural trapsWarm upWealth and advisoryDistribution and sales

    Try it first

    Your instinct, before any arithmetic.

    Show the worked solution

    52%, the same as on any other day. If each day's move is independent, the five falls carry no information about tomorrow. A run of five falls was unlikely before it began, 0.48 to the fifth power or about 2.5%, but that probability belonged to days that are now finished. Believing a rise is due is the gambler's fallacy.

    Why does the streak not make a rise more likely?

    A captain who has lost five tosses in a row walks out for the sixth feeling owed a win. The coin has no record of the first five and no sense of fairness to restore; it is still 50:50. Independence means past outcomes do not enter the calculation for the next one, so tomorrow's chance of a rise is the 52% it always was. The feeling that things must even out is real, and it has a name: the gambler's fallacy. It is strongest exactly when a streak is long, which is when it does the most damage to decisions.

    Five falls behind you, and tomorrow's odds have not movedDay 1fellDay 2fellDay 3fellDay 4fellDay 5fellDay 6: tomorrow52%chance of an up dayThe odds each morning, before the day's move: green up, red down52 / 4852 / 4852 / 4852 / 4852 / 4852 / 48, the same barBefore Monday: five falls in a row = 0.48 x 0.48 x 0.48 x 0.48 x 0.48 = 2.5%After Friday: that rare thing has already happened, and tomorrow is still 52%
    Each of the five days began with the same 52% chance of a rise, and so does tomorrow: the 2.5% chance of five falls in a row applied before the streak started, not after it has happened.

    Where does the tiny probability come from, and why is it the wrong number?

    It answers a different question. Standing on Monday morning, the chance of five falls in a row was 0.48 multiplied by itself five times, about 2.5%, and the chance of six in a row was about 1.2%, roughly one in 82. Once five falls have happened, the only uncertainty left is tomorrow, and for independent days the chance of a rise given the streak equals the chance of a rise on any day. Six falls in a row is rare only when viewed from the start; viewed from Friday evening, it needs just one more fall.

    The relationship
    P(up6∣5 downs)=0.485×0.520.485=0.52P(\text{up}_6 \mid \text{5 downs}) = \frac{0.48^5 \times 0.52}{0.48^5} = 0.52
    0.48^5the chance of the five falls that have already happened
    0.52the chance of a rise on any single day
    |read as given that
    What it says in wordsDivide the chance of the whole sequence by the chance of the part already seen, and the streak cancels out, leaving 52%.

    Is the independence assumption true for real markets?

    Not exactly, and saying so earns credit. Day-to-day direction in a broad index is very hard to predict from the previous days, but the size of moves does cluster: a run of falls often comes with bigger swings, so tomorrow's move may be larger even if its direction is a near coin toss. For an adviser the danger is not the arithmetic but the conversation: a client who believes a bounce is due will add money for the wrong reason, or hold out for a rebound the odds do not promise. A streak on its own is not a reason to act; any case for investing has to rest on the client's plan and horizon, not on the last five days.

    Where candidates lose it

    The trap is answering with the streak's rarity. Candidates say six falls in a row is roughly a one in 82 event, so a rise is almost certain, and in doing so treat days that are already over as if they were still uncertain. The interviewer is checking whether you can separate the probability of a sequence from the probability of the next step.

    The quieter loss is stopping at 52% without naming the assumption. Say independent, then add one sentence on what real markets do differently, such as swings clustering, and the answer sounds like someone who has watched markets rather than memorised a rule.

    What the interviewer asks next

    • What is the chance of at least one up day in the next five?
    • If an up day followed an up day 55% of the time, how would your answer change?
    • A client wants to invest only after a 5% fall. How would you explain the cost of waiting?
  8. 081Scheme A, with a NAV of Rs 42.00, is merged into scheme B, whose NAV is Rs 18.00. How many units of B does an investor holding 1,000 units of A receive, and has anyone gained or lost in the swap?NAV, units and fund mechanicsWarm upFund operationsRegistrars and transfer agents

    Try it first

    How many units of scheme B does the investor receive?

    Show the worked solution

    2,333.333 units, and nobody gains or loses on the day. The holding is worth 1,000 x Rs 42, or Rs 42,000. Divided by B's NAV of Rs 18, that is 2,333.333 units, and 2,333.333 x Rs 18 is Rs 42,000 again. The swap ratio, 42 over 18, is 2.333 units of B per unit of A. A merger changes the unit count and the scheme, not the value held.

    Why does a lower NAV not make scheme B cheaper?

    Change a Rs 500 note into Rs 100 notes and you get five of them. You now hold more notes, but you are no richer. A NAV is the value of one unit, so the number of units you hold only means something when multiplied by the NAV; a merger at NAV swaps notes of one size for notes of another. Scheme B's Rs 18 NAV only says its units started at a lower price or have grown less since launch. It says nothing about whether B is cheap, good or bad.

    The unit count changes; the value held does notUnits heldBefore: scheme A1,000 unitsAfter: scheme B2,333.333Value held, Rs1,000 x Rs 42.00Rs 42,0002,333.333 x Rs 18.00Rs 42,000Swap ratio = NAV of A / NAV of B = 42 / 18 = 2.333 units of B for every unit of A
    The investor's 1,000 units of A become 2,333.333 units of B, but both holdings are worth Rs 42,000, because the swap ratio of 2.333 is set by the two NAVs on the merger date.
    The relationship
    uB=uA×NAVANAVB=1,000×4218=2,333.333u_B = u_A \times \frac{NAV_A}{NAV_B} = 1{,}000 \times \frac{42}{18} = 2{,}333.333
    u_A, u_Bunits held in scheme A before and scheme B after
    NAV_A, NAV_Beach scheme's net asset value per unit on the merger date
    What it says in wordsThe new unit count is the old count times the ratio of the two NAVs, which keeps the rupee value unchanged.

    So can a merger leave an investor worse off?

    Not on the day of the swap, when both schemes are valued at their closing NAVs. What a merger can change is everything after the swap: a different portfolio, a different expense ratio, a different level of risk, and possibly a tax event. If scheme B charges more, holds riskier bonds or follows another strategy, the investor's future returns change even though the swap itself was fair. That is why investors in a scheme being merged are generally offered a window to exit without an exit load; confirm the current rules on how that window works before relying on it.

    The three decimals are not decoration. Registrars in India commonly allot units to three decimal places, so the answer is 2,333.333 and not a rounded 2,333. Rounding down to whole units would take about Rs 6 from this investor, and across lakhs of folios that adds up, so fractional units exist precisely to keep the swap exact.

    Where candidates lose it

    The fast wrong answer is 1,000 units, as if a merger were a change of name. It would leave the investor with Rs 18,000 in place of Rs 42,000. The next wrong answer inverts the ratio and gives about 429 units. Both come from working with unit counts instead of rupees.

    State the rupee value first, Rs 42,000, and divide by the new NAV. Then add the sentence that shows judgement: the swap is fair on the day, and the real question is what the investor now owns and what it costs.

    What the interviewer asks next

    • Scheme B's expense ratio is 0.5 points higher. Roughly what does that cost on Rs 42,000 over ten years?
    • In what circumstances could the merger be a taxable event for the investor, and what would you check?
    • The investor had a monthly systematic investment plan into scheme A. What should happen to it?
  9. 082Without a calculator: at 8% a year, how long does Rs 1 lakh take to double, and how close does the rule of 72 get to the exact answer? Then do the same at 6% and at 12%.Compounding and time valueWarm upIndian AMCsDistribution and sales

    Try it first

    How far is the rule of 72 from the exact doubling time at 8%?

    Show the worked solution

    About 9 years at 8%; the exact figure is 9.006 years, so the rule of 72 is out by about 2 days. At 6% the rule gives 12 years against an exact 11.90, and at 12% it gives 6 years against an exact 6.12. In the range where fund returns usually sit, the rule is off by weeks, not years.

    Why does 72 divided by the rate give the doubling time?

    If the price of a Rs 100 thali rises 8% a year, it costs about Rs 200 in nine years. You did not need a calculator, only the fact that 72 over 8 is 9. The exact doubling time is ln 2 divided by ln(1 + r), and for rates of a few per cent ln(1 + r) is close to r, so the answer is close to 69.3 divided by the rate in per cent. Using 72 instead of 69.3 does two jobs: it nudges the answer up to correct for rates around 8%, and it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, which is what makes it a mental tool.

    The relationship
    t=ln⁡2ln⁡(1+r)≈72100 r0.6931ln⁡1.08=9.006t = \frac{\ln 2}{\ln(1+r)} \approx \frac{72}{100\,r} \qquad \frac{0.6931}{\ln 1.08} = 9.006
    tyears for money to double
    rthe annual rate, as a decimal
    lnthe natural logarithm
    What it says in wordsThe exact doubling time is log 2 over log of one plus the rate, and 72 over the rate in per cent is a close, easy stand-in.
    Rule of 72 against the exact doubling time, in yearsAt 6% a yearrule is long by 38 days12.0011.90At 8% a yearrule is short by 2 days9.009.006At 12% a yearrule is short by 42 days6.006.12036912Years to doublerule of 72exact
    The rule of 72 gives 12, 9 and 6 years at 6%, 8% and 12%, against exact doubling times of 11.90, 9.006 and 6.12 years, so in the range fund returns occupy it is wrong by weeks at most.

    Where does the rule start to slip?

    At both ends, but slowly. At 6% it says 12 years and the truth is 11.90, so it is long by about 38 days. At 12% it says 6 years against 6.12, short by about 42 days. The rule is almost exact near 8% and drifts by a little over a month either side, which is far inside the uncertainty of any return assumption you would feed it. Only at high rates does the drift become worth mentioning: at 24% the rule says 3 years against an exact 3.22.

    Two uses in a fund interview. Turn a return into something a client can feel: at 8%, money doubles roughly every nine years, so it quadruples in about eighteen. And run a claim backwards to test it: a fund that says it tripled money in ten years has compounded at about 11.6% a year, because tripling is about 1.6 doublings, one every 6.3 years, and 72 over 6.3 is a little over 11.

    Where candidates lose it

    The common slip is dividing 100 by the rate, which gives 12.5 years at 8%. That is what simple interest would give, with no interest earned on interest, and it overstates the wait by about three and a half years.

    The quieter loss is presenting the rule as exact. Say 9 years, then add that the exact figure is 9.006, so the interviewer hears that you know what the shortcut is and where it stops working.

    What the interviewer asks next

    • How long does money take to triple at 8%?
    • Inflation runs at 6%. How long before Rs 1 lakh buys half what it buys today?
    • What number would you use in place of 72 for continuously compounded rates, and why?
  10. 085A fund's monthly returns have a standard deviation of 5%. What is that as an annual volatility, and is a month of minus 12% a rare event?Risk, volatility and drawdownWarm upRisk and complianceIndian AMCs

    Try it first

    What is the fund's annual volatility?

    Show the worked solution

    About 17.3% a year, and a minus 12% month is a 2.4 standard deviation event: rare, not freakish. Annual volatility is the monthly figure times the square root of 12: 5% x 3.46 is 17.3%. A minus 12% month is 12 divided by 5, or 2.4 monthly standard deviations. A normal curve puts that at about 0.8% of months, roughly once a decade, and real markets produce such months more often.

    Why does volatility scale with the square root of time?

    Take ten steps where each one goes left or right on a coin toss. You rarely end ten steps from where you began; lefts and rights cancel, and a typical distance is only about three steps, the square root of ten. Monthly returns behave the same way: good and bad months partly cancel, so variance adds up over time and the standard deviation grows only with the square root of the number of months. That is why the conversion uses the square root of 12, about 3.46, and not 12.

    The relationship
    σyear=σmonth12=5%×3.464=17.3%z=−12%5%=−2.4\sigma_{year} = \sigma_{month}\sqrt{12} = 5\% \times 3.464 = 17.3\% \qquad z = \frac{-12\%}{5\%} = -2.4
    \sigma_{month}the standard deviation of monthly returns, 5%
    \sqrt{12}the square root of the number of months in a year
    zhow many monthly standard deviations the fall is from an average month
    What it says in wordsScale volatility up by the square root of time, and measure a single month against the monthly figure, not the annual one.
    Monthly returns with a 5% standard deviation, and where minus 12% sits-15%-3 sd-10%-2 sd-5%-1 sd0%+5%+1 sd+10%+2 sd+15%+3 sdminus 12%= -12 / 5 = -2.4 sdShaded tail:0.82% of months, 1 in 122Annual volatility5% x the square root of 12= 17.3% a yearMonthly return
    With a monthly standard deviation of 5%, a minus 12% month lies 2.4 deviations below the centre, and the normal curve puts only 0.82% of months beyond it, about one in 122, while the same fund's annual volatility is 17.3%.

    So how rare is a minus 12% month?

    Measure it in monthly standard deviations. A minus 12% month is 2.4 monthly standard deviations below an average month taken as zero, which a normal curve puts at about 0.8% of months, roughly one in 122, or once a decade. If the fund's average month is plus 1%, the fall is 2.6 deviations and a normal curve makes it a little rarer, 0.47%. Against the 17.3% annual figure the fall sounds small, and that is the confusion to avoid: judge a month on a monthly scale.

    Now the limitation. Equity returns have fatter tails than the normal curve: large falls cluster in crises and turn up more often than the bell shape allows. Treat the normal figure as a floor on how often such a month arrives, not as a promise that it will be rare. For a client the useful sentence is plain: a fund with annual volatility near 17% will, from time to time, lose more than a tenth of its value in a single month.

    Where candidates lose it

    The first loss is scaling the wrong way. Multiplying by 12 gives 60%, which describes a far wilder fund than this one, and it comes from treating risk as if it added up like returns. Say variance adds, then take the square root.

    The second is judging the minus 12% month against the annual 17.3% and calling it ordinary. Put the move and the volatility on the same time scale before comparing them, and the month turns out to be a 2.4 deviation event.

    What the interviewer asks next

    • What is this fund's volatility over a single week?
    • The fund's worst month in ten years was minus 20%. What does that tell you about the normal assumption?
    • Why might reported annual volatility understate the risk of a bad month?